Various analytical methods employing X-rays are well-known. See, e.g., Van Nostrand's Scientific Encyclopedia, "X-ray Analysis," pp 2353-2356 (5th ed. 1976); Stern et al., X-rays, pp. 31-35 and 202-211 (1970); Frevert et al., "On-line Non-contacting Determination of Ash Content in Fast-moving Paper Webs," Industrial Measurement and Control by Radiation Techniques, pp. 208-214 (1972); and U.S. Pat. Nos. 4,047,029, 3,861,199, and 3,114,832.
In particular, X-ray fluorescence is well-known (see, e.g. Considine (editor), Process Instruments and Controls Handbook, "X-ray Fluorescence Analytical Methods," (1st ed. 1957) and "X-ray Fluorescence Analysis,"(2nd ed. 1974), and U.S. Pat. No. 4,169,228) and has been used in the paper industry. See, e.g., Puumalalnen et al., "A new moisture-insensitive method for measurement of paper and board coating weights," TAPPI, vol. 63, no. 7, pp. 55-57 (1980); Buchnea et al., "On-line Non-destructive Paper Chemistry Analysis by X-ray Fluorescence," Am. Nucl. Soc. Trans., vol. 22, pp. 146-148 (1975); McNelles et al., "An On-line Ash Constituent Determination Using X-ray Fluorescence"; and U.S. Pat. Nos. 4,081,676 and 3,660,662.
This technique is based on the discovery years ago that if atoms of an element are excited by photons of sufficient energy (primary beam), those atoms will give off photons having energy characteristic of that element (fluorescent or secondary radiation). In practice, however, analyzing the fluorescent radiation data to determine how much of a particular element is present in a sample is made complex by what may be called the "position effect" and by the so-called "matrix effect," which significantly influence the data.
The position effect arises when the element of interest is not uniformly distributed throughout the sample, that is, when there is a concentration gradient. Various techniques have been suggested for eliminating the position effect. See, e.g., U.S.S.R. Pat. No. 491,883 and Vander, "Method of Measurement of Mean Concentration for an Element Segregated in Layers by X-ray Analysis," Advances in X-ray Analysis, vol. 21, pp. 143-147 (1978). They suggest that the fluorescence be measured from the non-irradiated side of the sample and that the apparatus be arranged to satisfy the equation ##EQU2## where .mu..sub.M,1 and .mu..sub.M,2 are the mass absorption coefficients of the matrix for the incident (or primary) beam and for the fluorescent beam, respectively, and .PSI. and .PHI. are the angles of the fluorescent and incident beams, respectively, to the sample.
Of greater significance, however, is the matrix effect, which occurs because of the presence of elements in the sample in addition to the element of interest. The other elements may cause the concentration of the element of interest calculated from the test data to be significantly higher or lower than the actual concentration.
Numerous techniques have been suggested to compensate for the matrix effect. Some involve iterative solution of simultaneous equations in which the concentration of each element is an unknown. This, in turn, requires the use of a computer. (Even then, the solution may not be mathematically stable.) See, e.g., Lucas-Tooth et al., "The Accurate Determination of Major Constituents by X-ray Fluorescent Analysis in the Presence of Large Interelement Effects," Advances in X-ray Analysis, vol. 7, pp. 523-541 (1964); and Lucas-Tooth et al., "A Mathematical Method for the Investigation of Inter-Element Effects in X-ray Fluorescent Analysis," Metallurgia, vol. 64, pp. 149-152 (1961).
Another technique for compensating for the matrix effect is disclosed in U.S.S.R. Pat. No. 171,482. This method utilizes the ratio between fluorescent and scattering radiation; however, accurate discrimination between fluorescent photons and scattered photons having energies close to the fluorescent photons is difficult.
Another technique that involves matrix compensation when assaying for three specific components in a web such as paper is disclosed in U.S. Pat. No. 4,081,676. Various assumptions are made initially and expected absorption is compared to actual absorption.
Yet another method involving matrix compensation when assaying for four specific components in paper is disclosed in Puumalalnen et al., above. Absorption data and fluorescence data from both sides of the sample are utilized in an iterative procedure.
Each of these matrix compensation techniques has drawbacks.